Geometric quantization and internal symmetry

As a part of an attempt to geometrize physics, internal symmetries in the covariant classification of matter by itsT type are considered in relation to phase transformations generated by complex and quaternionic structures on space-time. The Rainich theory of electromagnetism and neutrinos is compared with the theory ofU(1) ×SO(1, 3) torsional gauge fields, and extended to the quaternionic case. It is shown by the Kostant technique of geometric quantization that complex and quaternionic phase transformations for an Einstein space are associated with one-dimensional and three-dimensional harmonic oscillators.     

Nonlinear Dirac Operator and Quaternionic Analysis

Properties of the Cauchy–Riemann–Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3–surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions of the Cauchy–Riemann–Fueter equation are established.     

Tensors,spinors, and functions on the unit sphere

A representation of tensors and spinors at a point of space-time as spin and conformally weighted functions on the unit sphere is derived. Methods for performing algebraic operations on tensors and spinors in this representation are discussed.Supported in part by grants Nos. GP-35773X1 and MPS74-18020 from the National Science Foundation.     

Null vectors,spinors, and strings

It is shown how, in the frame of the Cartan-conception of spinors, the old theorems onminimal surfaces, as generated from null-curves, formulated by Enneper-Weierstrass (1864–1866) for 3-dimensional ordinary space, and by Eisenhart (1911) for 4-dimensional space-time, may be reformulated in terms ofcomplex 2- and 4-component projective spinors respectively. For the correspondingreal (Majorana) spinors instead the same procedure naturally leads tostrings in 3-dimensional and 4-dimensional space-time (?^{2, 1} and ?^{3, 1}). It is suggested that this close connection with Cartan-spinors, and the corresponding (projective) null-geometry, may be the clue for understanding the fundamental nature of strings.     

Charged Representations of the Infinite Fermi and Clifford Algebras

The real and quaternionic charge conjugation operators invariant under the infinite-dimensional Clifford algebra, or compatible with the Fermi algebra, are determined. There results a maze of inequivalent irreducible charged representations, all of which are non-Fock. The representation vectors and their charges admit two interpretations besides those of spinors or states of quantum fields: as wavelets on the circle, with charge conjugations acting via ordinary complex conjugation; and as infinite-dimensional numbers, with charge conjugations acting by automorphisms.     

Octonions and the Lorentz and conformal groups of ten-dimensional space-time

Replacing complex numbers by octonions enables spinors and twistors to be defined for ten-dimensional space-time in close analogy to the four-dimensional case.     

Pure spinors and quadric Grassmanians

If, following E. Cartan, the simplest spinors (pure) are conceived as equivalent to isotropic (or null or optical) polarized planes in complex spaces, then the most natural tensors generated (bilinearly) by the simplest spinors are isotropic vectors rather than ordinary linear ones. The conjecture that spinors are fundamental would then imply that non-linear geometry of isotropic elements should be more elementary in general than the linear one; and the relevance of optical geometry (optical flags, optical groups) on space-time manifolds for the explanation of optical phenomenology in the frame of general relativity [5] could already constitute a first confirmation of this conjecture.Only 2- and 4-component spinors build up linear spinor spaces while 8, 16, 32,...component pure spinors, instead, are subject to covariant (quadratic) constraint equations and build up non-linear sets isomorphic, up to a sign, to quadric Grassmanians and, for neutral and conformal spaces, to Lie groups.The possible relevance of such pure spinor properties for physics is conjectured and exemplified.     

Particles,twistors and the division algebras

We study twistorial mechanics of particles and super-particles in six dimensions. To this end we formulate (in a general division algebra framework) a twistor theory in D = 6 based on quaternionic numbers, and prove the equivalence between this version of particle dynamics and the ordinary one. The super-twistors define a covariant and gauge invariant concept of a super world-line and allow us to write an action for the supersymmetric particle that is not plagued by the content of second class constraints that prevents a covariant quantization in the space-time picture. The notion and geometry of projectile twistor space, and its connection to Minkowski space, are examined and shown to directly generalize the results in D = 3, 4. Quantization is performed and analytic quaternionic eigenfunctions and integrations are discussed. We also draw some conclusions on the possible generalization to ten dimensions.     

Cartan mapping and spinors in affinely connected space-time

It is shown that Cartan spinors with a pair of mutual spinor connections exist only in a Weyl-nonmetric, affinely connected space-time with a torsion tensor absolutely antisymmetric in an orthonormal frame.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 84–88, January, 1995.     

Spin-3/2 fields in flat space-time

The equations for the spin-3/2 (Rarita-Schwinger) field given by linearized simple supergravity are written in space-plus-time form in terms ofSU(2) spinors, assuming that the background space-time is flat. Some consequences of these equations are analyzed and a Hamiltonian structure for the Rarita-Schwinger field is obtained.     

Soler's Theorem and Characterization of Inner Product Spaces

Using Soler's result, we show that the existenceof at least one finitely additive probability measure onthe system of all orthogonally closed subspaces of Swhich is concentrated on a one-dimensional subspace of E can imply that E is a real,complex, or quaternionic Hilbert space. In addition,using the concept of test spaces of Foulis and Randalland introducing various systems of subspaces of E , we give some characterizations of inner productspaces which imply that E is a real, complex, orquaternionic Hilbert space.     

Transmission resonance for a Dirac particle in a one-dimensional Hulthén potential

We have solved exactly the two-component Dirac equation in the presence of a spatially one-dimensional Hulthén potential, and presented the Dirac spinors of scattering states in terms of hypergeometric functions. We have derived the reflection and transmission coefficients using the matching condition on the wavefunctions, and investigated the condition for the existence of transmission resonance. Furthermore, we have demonstrated how the transmission resonance depends on the shape of the potential.      

Spinors,algebraic geometry,and the classification of second-order symmetric tensors in general relativity

Two approaches to the problem of classifying second-order symmetric tensors in space-time given by Ludwig and Scanlon and by Penrose are discussed. Ludwig and Scanlon use both spinor and tensor algebra in their approach, whereas Penrose uses spinors and the properties of certain curves in complex projective 3-space. These approaches yield essentially identical classifications, and this paper points out the connections between them in detail and tabulates the results.     

A new construction for spinor wave equations

The construction of discrete scalar wave propagation equations in arbitrary inhomogeneous media was recently achieved by using elementary dynamical processes realizing a discrete counterpart of the Huygens principle. In this paper, we generalize this approach to spinor wave propagation. Although the construction can be formulated on a discrete lattice of any dimension, for simplicity we focus on spinors living in 1+1 space-time dimensions. The Dirac equation in the Majorana-Weyl representation is directly recovered by incorporating appropriate symmetries of the elementary processes. The Dirac equation in the standard representation is also obtained by using its relationship with the Majorana-Weyl representation. Received: 3 November 1997 / Received in final form: 9 February 1998 / Accepted: 16 February 1998     

Production of Dirac particles due to Riccion coupling

It has been noted that at high energy the Ricci scalar is manifested in two different ways, as a matter field as well as a geometrical field (which is its usual nature even at low energy). Here, using the material aspect of the Ricci scalar, its interaction with Dirac spinors is considered in four-dimensional curved space-time. We find that a large number of fermion-antifermion pairs can be produced by the exponential expansion of the early universe.     

A lagrangian formulation of high-spin fields coupled to gravity

A coupling of high (half-integer) spin fields to gravity is discussed in terms of symmetric tensor-valued spinors for which restrictive integrability constraints are avoided. The theory is generated from a spin-invariant action which provides a means of computing the associated stress tensor. Explicit no-ghost solutions are presented and a covariance of the system under a local conformai isometry group of space-time is pointed out.We are grateful to G. R. Allcock for helpful correspondence.     

Spin-orbit lateral superlattices: Energy bands and spin polarization in 2DEG

The Bloch spinors, energy spectrum, and spin density in energy bands are studied for a two-dimensional electron gas (2DEG) with Rashba spin-orbit (SO) interaction subject to the one-dimensional (1D) periodic electrostatic potential of a lateral superlattice. The space symmetry of the Bloch spinors with spin parity is studied. It is shown that the Bloch spinors at fixed quasi-momentum describe the standing spin waves with the wavelength equal to the superlattice period. The spin projections in these states have components both parallel and transverse to the 2DEG plane. The anticrossing of the energy dispersion curves due to the interplay between the SO and periodic terms is observed, thus, leading to the spin flip. The relation between the spin parity and the interband optical selection rules is discussed, and the effect of magnetization of the SO superlattice in the presence of an external electric field is predicted. The text was submitted by the authors in English.     

4d fermionic superstrings with arbitrary twists

We present the rules for systematically constructing all consistent four-dimensional string theories, using free world-sheet fermions which pick up arbitrary phases when parallel transported around the string. These rules are necessary and sufficient for multi-loop modular invariance. They lead to theories with general Z_N (GSO-type) projections, whose merits for model-building we discuss. We classify all boundary conditions yielding massless space-time spinors. We show that, in contrast to the case of only real 2d fermions, all possible realizations of world-sheet supersymmetry are now allowed. This opens the way for the construction of a new class of supersymmetric string models.     

Spinor algebra of the one-component lattice fermions

We develop a general formalism to translate the Susskind one-component theory of free fermions in a formulation with conventional Dirac spinors. It results that Susskind theory has parity-violating and flavour-changing terms which vanish only in the continuum limit. Gauge fields destroy the equivalence between the one-component and the conventional formulations. In particular the one-component fermions respond to a gauge field as they were in a curved space-time. However, this gravity effect vanishes if one takes a naive continuum limit.